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(A) Given determinant is $\Delta = \left|\begin{array}{ccc} \log e & \log e^2 & \log e^3 \ \log e^2 & \log e^3 & \log e^4 \ \log e^3 & \log e^4 & \l

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(A) Given determinant is $\Delta = \left|\begin{array}{ccc} \log e & \log e^2 & \log e^3 \ \log e^2 & \log e^3 & \log e^4 \ \log e^3 & \log e^4 & \log e^5 \end{array}ight|$.Using the property $\log a^n = n \log a$,we can rewrite the determinant as:$\Delta = \left|\begin{array}{ccc} \log e & 2 \log e & 3 \log e \ 2 \log e & 3 \log e & 4 \log e \ 3 \log e & 4 \log e & 5 \log e \end{array}ight|$.Taking $\log e$ common from each column,we get:$\Delta = (\log e)^3 \left|\begin{array}{ccc} 1 & 2 & 3 \ 2 & 3 & 4 \ 3 & 4 & 5 \end{array}ight|$.Applying column operations $C_2 ightarrow C_2 - C_1$ and $C_3 ightarrow C_3 - C_2$:$\Delta = (\log e)^3 \left|\begin{array}{ccc} 1 & 1 & 1 \ 2 & 1 & 1 \ 3 & 1 & 1 \end{array}ight|$.Since column $C_2$ and column $C_3$ are identical,the value of the determinant is $0$.

$\left|\begin{array}{ccc} \log e & \log e^2 & \log e^3 \\ \log e^2 & \log e^3 & \log e^4 \\ \log e^3 & \log e^4 & \log e^5 \end{array}\right| \text{ is equal to: }$

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